Abstract
AbstractLet D be a connected bounded open set in
$$\mathbb {R}^d$$
ℝ
d
and let v ∈ H1(D) be a given non-negative function. This chapter is dedicated to the problem
$$\displaystyle \begin{aligned} \min\Big\{\mathcal F_0(u,D)\ :\ u\in H^1(D),\ u-v\in H^1_0(D),\ |\Omega_u\cap D|=m\Big\}, \end{aligned} $$
min
{
ℱ
0
(
u
,
D
)
:
u
∈
H
1
(
D
)
,
u
−
v
∈
H
0
1
(
D
)
,
|
Ω
u
∩
D
|
=
m
}
,
where m ∈ (0, |D|) is a fixed constant and we recall that
$$\displaystyle \begin{aligned} \mathcal F_0(u,D)=\int_D|\nabla u|{}^2\,dx. \end{aligned}$$
ℱ
0
(
u
,
D
)
=
∫
D
|
∇
u
|
2
d
x
.
In this chapter, we give the main steps of the proof of Theorem 1.9.
Publisher
Springer International Publishing
Reference6 articles.
1. F.J. Almgren Jr., Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents. Minimal Submanifolds and Geodesics (Kaigai, Tokyo, 1979), pp. 1–6
2. T. Briançon, Regularity of optimal shapes for the Dirichlet’s energy with volume constraint. ESAIM, Control Optim. Calc. Var. 10, 99–122 (2004)
3. T. Briançon, J. Lamboley, Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(4), 1149–1163 (2009)
4. G. Buttazzo, B. Velichkov, A shape optimal control problem with changing sign data. SIAM J. Math. Anal. 50(3), 2608–2627 (2018)
5. N. Garofalo, F.-H. Lin, Monotonicity properties of variational integrals, A
p weights and unique continuation. Indiana Univ. Math. J. 35, 245–268 (1986)